Summary: Journal of Algebraic Combinatorics 10 (1999), 207­225
c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
Extended Linial Hyperplane Arrangements for Root
Systems and a Conjecture of Postnikov and Stanley
CHRISTOS A. ATHANASIADIS athana@math.upenn.edu
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395
Received September 2, 1997
Abstract. A hyperplane arrangement is said to satisfy the "Riemann hypothesis" if all roots of its characteristic
polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families
of arrangements which are defined for any irreducible root system and was proved for the root system An-1.
The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent
combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar
formulae for all but the exceptional root systems. The conjecture follows in these cases.
Keywords: hyperplane arrangement, characteristic polynomial, root system
1. Introduction
Let A be a hyperplane arrangement in Rn
, i.e. a finite collection of affine subspaces of Rn
of codimension one. The characteristic polynomial [9, §2.3] of A is defined as
(A, q) =
xLA